How do you prove the area of a circle?
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This result can be proven using A-level integration. The idea is to first form the area for a small segment of a circle, with a small angle d(theta). For a very small angle this is approximately a triangle, and so it's area is (1/2) base x height. The height is the radius r, and the base is the arc length s= r*d(theta) (another result that is learnt in A-level maths). In other words, the area of this segment is: dA= (1/2) r^2 d(theta) You then integrate over the whole circle 0 < theta < 2pi to find the full area: A = (1/2) r^2 2pi = pi*r^2 and the result is proven.
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If the object is a circle, and you know its circumference, you would divide the circumference by pi to find the diameter of the circle. Half the diameter is the radius. Square the radius and multiply by pi to find the area of the circle.
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